Question

1. the following mosaic plot displays the information gathered in a recent statistics survey, that asked

the class what their favorite subject is. the data was recorded, along with their gender.
which of the following interpretations, based on the mosaic plot, is false?
(a) english is the least popular subject among all students.
(b) 83% of those who listed english as their favorite subject were male.
(c) more males than females listed science as their favorite subject.
(d) more females than males listed math as their favorite subject.
(e) the number of males that listed history as their favorite subject is equal to the number of males who listed
art/music as their favorite subject.

1. an economist in a large city wanted to develop a model to estimate the vacancy rate of downtown

office space (the percent of available office space that was unoccupied) based on the monthly rental price per
square foot. she took a random sample of various office buildings in the downtown area and created three
models using monthly rental prices (x) to predict vacancy rate (y).
i. a model using (x, y) produced a regression equation of \\( \hat { y } = 1.176 x - 3.019 \\) and the residual plot
showed strong curvature.
ii. a model using \\( ( x , sqrt { y } ) \\) produced a regression equation of \\( sqrt { hat { y } } = 0.157 x + 1.3497 \\) and the residual plo
indicated that the variance increased as the rental prices increased (a \fanning\ effect).
iii. a model using \\( ( x , ln ( y ) ) \\) produced a regression equation of \\( ln ( hat { y } ) = 0.038 x + 0.538 \\) and the resic
plot showed no apparent pattern.
based on the above information, which transformation(s) would be most appropriate to linearly model th
(a) i only
(b) ii only
(c) iii only
(d) i and ii
(e) none of the above would be appropriate

Explanation:

Step1: Analyze Model I

A regression equation \(\hat{y}=1.176x - 3.019\) with a residual plot showing strong curvature. Strong curvature in the residual plot indicates that the linear model (Model I) is not appropriate as it does not fit the data well.

Step2: Analyze Model II

A model using \((x,\sqrt{y})\) with a regression equation \(\sqrt{\hat{y}}=0.157x + 1.3497\) and a residual plot showing increased variance (fanning effect). Unequal variance (heteroscedasticity, shown by fanning) violates the assumptions of linear regression, so Model II is not appropriate.

Step3: Analyze Model III

A model using \((x,\ln(y))\) with a regression equation \(\ln(\hat{y})=0.038x + 0.538\) and a residual plot showing no apparent pattern. A residual plot with no pattern is a good sign for a linear model. When we transform \(y\) to \(\ln(y)\) and the residual plot has no pattern, it indicates that the transformed model (Model III) is appropriate for linear modeling.

Answer:

C. III only